Download Waves & Oscillations Physics 42200 Spring 2015 Semester

Physics 42200
Waves & Oscillations
Lecture 25 – Introduction to Optics
Spring 2015 Semester
Matthew Jones
Introduction to Optics
• If we can identify any system that is described by
the wave equation, then we know everything!
1
= • Except that…
–
–
–
–
The geometry might be complicated
Boundary conditions might not be simple
Initial conditions might not be known
The solution is the superposition of incident waves and
all possible reflected waves
• So we develop new ways of looking at this problem
and call it optics.
Electromagnetism
• Geometric optics overlooks the wave nature of
light.
– Light inconsistent with longitudinal waves in an
ethereal medium
– Still an excellent approximation when feature sizes are
large compared with the wavelength of light
• But geometric optics could not explain
– Polarization
– Diffraction
– Interference
• A unified picture was provided by Maxwell c. 1864
Forces on Charges
• Coulomb’s law of electrostatic force:
$#
Charles-Augustin de
Coulomb
(1736 - 1806)
$
• The magnitude of the attractive/repulsive force is
where
= ̂
1
=
= 8.99 × 10 ∙ ∙
4
and therefore
!
= 8.85 × 10!# ∙ !# ∙ !
(This constant is called the “permittivity of free space”)
Electric Field
• An electric charge changes the properties of the
space around it.
– It is the source of an “electric field”.
– It could be defined as the “force per unit charge”:
# = 5%
%=
̂
&'() – Quantum field theory provides a deeper description…
• Gauss’s Law:
1
$012034
*,
+ ∙ %-. =
/
Electric “flux” through surface S
Magnetic Field
• Moving charges (ie, electric current) produce a
magnetic field:
7 8-ℓ × ̂
*
6=
4
A moving charge in a magnetic field
experiences a force:
= 5
× 6
Lorentz force law:
= 5% + 5
× 6
Gauss’s Law for Magnetism
• Electric charges produce electric fields:
$012034
, ∙ %-. =
*+
/
• But there are no “magnetic charges”:
2
, ∙ 6-. =
*+
/
0
Magnetic “flux” through surface S
Ampere’s Law
• An electric current produces a magnetic field that
curls around it:
;
;
(Close, but not quite the whole story)
Faraday’s Law of Magnetic Induction
• Magnetic flux: <= = >/ 6 ∙ +? -.
• Faraday observed that a changing
magnetic flux through a wire loop
induced a current
– It transferred energy to the charge
carriers in the wire
-<=
ℇ = −
-
B % ∙ -ℓ = − * +? ∙ 6-.
- /
;
3
The Problem with Ampere’s Law
Current through C# is 8
Current through C is 0!
The current through C is zero,
but the electric flux is not zero.
The electric flux changes as
charge flows onto the capacitor.
Maxwell’s Displacement Current
• We can think of the changing electric flux
through C as if it were a current:
-<4
83 = = * % ∙ +? -.
-
- /
B D ∙ Eℓ = GH I + IE
F
E
, EQ
= GH J + GH KH * M ∙ N
EL OP
4
Maxwell’s Equations (1864)
RSNTSEU
, ∙ M EQ = B N
KH
O
, ∙ D EQ = H
B N
O
EVW
B M ∙ Eℓ = −
EL
F
EVU
B D ∙ Eℓ = GH I + GH XH
EL
F
1
2
3
4
Maxwell’s Equations in Free Space
In “free space” where there are no electric charges or
sources of current, Maxwell’s equations are quite
symmetric:
, ∙ M EQ = H
B N
O
, ∙ D EQ = H
B N
O
EVW
B M ∙ Eℓ = −
EL
F
EVU
B D ∙ Eℓ = GH XH
EL
F
Maxwell’s Equations in Free Space
EVW
B M ∙ Eℓ = −
EL
F
EVU
B D ∙ Eℓ = GH XH
EL
F
A changing magnetic flux induces an electric field.
A changing electric flux induces a magnetic field.
Will this process continue indefinitely?
Light is an Electromagnetic Wave
• Faraday’s Law:
∮; % ∙ -ℓ =
3Z[
−
3\
=−
]^_
]\
∆a∆b
B % ∙ -ℓ = %c b Δa − %c (b# )Δa
;
≈
x
]hi
∆b∆a
]j
Ex
z1
y
By
Δa
Ex
Δb
z2
z
Light is an Electromagnetic Wave
• Ampere’s law:
∮; 6 ∙ -ℓ =
3Zk
7 3\
=
]hi
7 ∆l∆b
]\
B 6 ∙ -ℓ = 6m b# Δl − 6m (b )Δl
;
≈−
]^_
]j
x
∆b∆l
Ex
∆b
z1 z2
y
By
By
∆l
z
Putting these together…
6m
%c
=−
b
6m
%c
−
= 7 b
Differentiate the first with respect to b:
6m
%c
=−
b
b
Differentiate the second with respect to :
6m
%c
−
= 7 b
nP M o
nP Mo
= GH KH
P
np
nLP
Velocity of Electromagnetic Waves
l
1 l
= a
nP M o
nP M o
= GH KH
P
np
nLP
Speed of wave propagation is
1
=
7 1
=
4 × 10!q /. 8.854 × 10!# = P. sst × uHt v/w
(speed of light)
/
∙
Light is an Electromagnetic Wave
• Maxwell showed that these equations contain the wave
equation:
1 l
l
= a
where l a, is a function of position and time.
• General solution:
l a, = y# a − + y (a + )
• Harmonic solutions:
l a, = l sin a ± ~
• Wavelength,  = 2/ and frequency y = ~/2.
• Velocity, = ⁄y = ~⁄.
The electromagnetic spectrum
• In 1850, the only known
forms electromagnetic
waves were ultraviolet,
visible and infrared.
• The human eye is only
sensitive to a very
narrow range of
wavelengths:
Discovery of Radio Waves
The Electromagnetic Spectrum
Electromagnetic Waves
nP Mo
nP Mo
= GH K H
P
np
nLP
• A solution is %c b, = % sin b − ~
where ~ = ‚ = 2‚/
• What is the magnetic field?
6m
%c
=−
= −% cos b − ~
b
6m a, = % sin a − ~
~
Electromagnetic Waves
ax
bz
ly
• %, 6 and are mutually perpendicular.
• In general, the direction is
…̂ = %† × 6†
Energy in Electromagnetic Waves
Energy stored in electric and magnetic fields:
1
1 ‡4 = %
‡= =
6
2
27
For an electromagnetic wave, 6 = % ⁄‚ = % 7 1 1
‡= =
6 = % = ‡4
27
2
The total energy density is
‡ = ‡= + ‡4 = % Intensity of Electromagnetic Waves
• Intensity is defined as the average power
transmitted per unit area.
Intensity = Energy density × wave velocity
%
8 = ‚ % =
7 ‚
7 ‚ = 377Ω ≡ Œ
(Impedance of free space)
Poynting Vector
• We can construct a vector from the intensity and the
direction …̂ = %† × 6†:
%×6
C
=
7
%
C
=
=8
Œ
• This represents the flow of power in the direction …̂
• Average electric field: %=2 = % / 2
% C
=
2Œ
• Units: Watts/m2
Have a nice spring break!